Outlines of Methodological Pieces

Outlines of Methodological Pieces

Cobb, P., Stephan, M., McClain, K., & Gravemeijer, K. (2001). Participating in classroom mathematical practices. Journal of the Learning Sciences, 10, 113-163.

INTRODUCTION (pp 113-115)

Begins by pointing to a call from Barab for more methodologies. This article is an attempt to answer that call with respect to collective mathematical learning; in particular, they try to blend a description (from Saxe) of methodology as data collection/analysis technique and methodology as questioning framework with epistemological assumptions (p 114). They then finish the introduction by giving a sentence or so description of each of the sections below.

Our article: I don't think we have such a specific impetus for the methodology, plus we aren't really developing it--more like reintroducing it. I also think that our methodology is focused more on the first interpretation of the term "methodology" (i.e., data collection and analysis technique) and the second aspects (i.e., framework) comes from elsewhere (i.e., systemic functional linguistics).


DESIGN RESEARCH (pp 115-118)

They have been doing research like this for 12 years, and they classify it as "design research" because "it involves both instructional design and classroom-based research" (p 115). The design research cycle is displayed on page 116. There instructional aspect is based on RME from Freudenthal, and the research portion is based on testing and modifying conjectures about trajectories of mathematical learning. They identify three "criteria that an analytical approach should satisfy if it is to enable us to contribute to reform in mathematics education as an ongoing, iterative process of improvement" (p 116):
(1) It should enable us to document the collective mathematical development of the classroom community over the extended periods of time covered by instructional sequences.
(2) It should enable us to document the developing mathematical reasoning of individual students as they participate in the practices of the classroom community.
(3) It should result in analyses that feed back to inform the improvement of our instructional designs.

They take a paragraph to unpack each of these criterion. Then they point out that design research rejects the top-down model of research and practice interaction, instead holding research and practice on equal footing.

Our article: This section basically outlines the things that they feel a methodology should offer to the field. They are also implicitly making the case for the need and value of design research. If we wanted, we could frontload the things that we eventually get out of the method, but it doesn't feel too natural for us. It also doesn't feel like we need to make a general case for discourse analysis, do we?


INTERPRETATIVE FRAMEWORK (pp 118-122)

Cobb et al used an interpretative framework to guide their analysis. It is built around the coordination of two dimensions (see p 119 for a table):
SOCIAL (aspects of classroom microculture) -- classroom social norms, sociomathematical norms, classroom mathematical practices.
PSYCHOLOGICAL (aspects of individual student activity) -- beliefs about roles and the nature of mathematical activity, mathematical beliefs and values, mathematical interpretations and reasoning.

They claim that these two perspectives address the first two criteria from above.

The authors trace the lineage of the social perspective (e.g., sociocultural theory) and the psychological (e.g., Piaget, Hutchins). "In summary, there is an extremely strong relation between what we have described as the social and psychological perspectives that does not merely mean that the two perspectives are interdependent. Instead, it implies that neither perspective exists without the other in that each perspective constitutes the background against which mathematical activity is interpreted from the other perspective" (p 122).

Our article: This section presents the guiding framework of their analysis. Again, it seems like they have more to do because they are almost constructing a methodology or introducing it from the bottom up. For us, it doesn't seem like we have to do so much. However, the visual guide to the underpinnings of the methodology was nice. Could we do (read: borrow) something like that?


ASPECTS OF THE CLASSROOM MICROCULTURE AND INDIVIDUAL STUDENTS' REASONING (pp 122-126)

The previous section (of which this is actually a subsection) gave the general underpinnings of their framework, and here they attempt to delve into the specifics. They basically use this space to give lengthy definitions and examples of the terms used in the framework. They reiterate that the framework is a guide for analysis, not a source of prescription for practice.

Our article: I'm sure we will need to define terms, but I don't know if it requires a separate section (or subsection). It seems as though we could roll it into the background section. (Or we could just say "see Lemke for definitions" and we're done, except as modifications come up like "extent.")


METHODOLOGICAL CONSIDERATIONS (pp 126-131)

"Our primary focus when we present the sample analysis is on the evolution of classroom mathematical practices, as this is the least developed aspect of the interpretive framework. Our unit of analysis, therefore, is that of a classroom mathematical practice and students’ diverse ways of participating in and contributing to its constitution" (p 126). They note that their corpus of classroom videos is massive, and so it is important to develop a systematic way to analyze the corpus. They rely on Glaser and Strauss to work from the data but make some modifications (e.g., they had some categories beforehand). The method has several phases. First, they work chronologically through the episodes and write memos of conjectures, revisions, and refutations. Second, these memos become data for meta-analysis and a chronology of the "mathematical learning of the classroom community" (p 128) is generated. Third, the results of this analysis are cast in terms of framework.

On page 129, they differentiate between types of mathematical norms. They then talk about what they use as warrants for making claims about these norms and sociomathematical norms.

Our article: Your project also has a massive video corpus, but our piece is in no way concerned with the entire thing. We've just chosen a couple small pieces to highlight what the methodology can offer. As for warrants, since we are focused on a short transcript, we can build the thematic map right from the lines of the transcript in full view of the reader.


MEASUREMENT PRACTICES (pp 131-146)

This basically begins by describing the participants, the background of the teaching experience (pp 131-133) and the classroom microculture (pp 133-134). They then describe the emergent mathematical practices -- measuring by pacing and measuring by iterating a footstrip (pp 135-137) -- and how they came about. Beginning on page 138 they describe in detail, with a transcript excerpt, pictures, and an in-paragraph chronology, the third mathematical practice of measuring by iterating a Smurf bar (p 142). They particularly focus on one student's reasoning around this practice. They then point out how the factors of this analysis effected their instructional design in the design research cycle (p 145).

Our article: For us, the first part will be found in the section that I called "project background." Then they are presenting their findings, which is not only showing the findings but more importantly giving insight into the method and what it takes into account. This will correspond to our development of the thematic maps from the transcripts, where we have some modest findings but are primarily demonstrating the methodology at work.


METHODOLOGICAL REFLECTIONS (pp 146-151)

They describe what they mean by "episode" in the analysis - a mathematical activity with a single theme as the focus - and look back at some examples (p 146). "Critical episodes" (in the second phase of analysis) are those that confirm or refute an assertion; thus they are critical with respect to the conjectures, not just on their own. They then try to describe how they delineated the mathematical practices (p 147), which differed from simply identifying solution strategies because they wanted to look at things "taken as shared". Third, they discuss "what is involved in taking a social perspective" (p 148). Fourth, they explain how they treated the use of tools in their analysis (p 149). On page 150 they describe why they rejected the epistemic fidelity view of tools (b/c it ignores students' prior experiences and the taken-as-shared purposes), but they don't abandon the notion of affordances. "As a final observation about our treatment of tool use, the reader may have noticed that we did not follow the standard sociocultural approach when speaking of students appropriating ways of reasoning with tools" (p 151).

Our article: I didn't feel like this section was as "reflective" as the title suggests; it seems more like they were just saying or clarifying some things that they couldn't fit elsewhere. Anyway, I can't think of exactly what we would do that is similar to this. It seems as though we would explain how we chose things and what we selected for analysis before we've actually presented the analysis. Also, I expect that we would explain how we're going to talk about things before we do the talking, rather than afterward like they did.


TRUSTWORTHINESS, REPLICABILITY, AND COMMENSURABILITY (pp 151-154)

Cobb et al point out that most interpretative analyses base their assertions on the presentation of a limited number of examples. But then the interpretations sometimes don't seem justified. "The most important criterion in this regard is the extent to which the analysis of a longitudinal data set of this type generated during a teaching experiment is both systematic and thorough. The hallmark of an analytical approach that satisfies this criterion is that inferences are treated as provisional conjectures that are continually open to refutation" (p 152). To this end, they tried to give the sense of this open process when writing and they also documented all stages of their analysis and conjectures; thus it is open for backtracking. They also write that having other researchers (some familiar with the project, some not) critique the analysis, and by having a prolonged relationship with the participant increases trustworthiness.

Regarding replicability, "[i]n contrast to traditional experimental research, the challenge as we see it is not that of replicating instructional treatments by ensuring that instructional sequences are enacted in exactly the same way in different classrooms. The conception of teachers as professionals who continually adjust their plans on the basis of ongoing assessments of their students’ reasoning would in fact suggest that complete replicability is neither desirable nor, perhaps, possible (Ball, 1993; Carpenter & Franke, 1998; Gravemeijer, 1994b). The challenge for us is instead to develop ways of analyzing treatments so that their realizations in different classrooms can be made commensurable" (p 153).

Our article: They talk about what validity looks like in this kind of research and make a case for why the method can be trusted. They also talk about how replicability is somewhat meaningless in the context and instead talk about commensurability (see pp 153-154). I think both of these would be useful for us to address in some way.


USEFULNESS/LIMITATIONS (pp 154-157)

They make three points with regard to usefulness. First, the approach documents the learning trajectory of the classroom community and at the same time it tracks the emergence of the "mathematical content" (p 154). It also answers two questions from Roth: "How does being in the world change in the course of activity? What are the long-term effects of students’ engagement in individual instructional activities?" Second, the method ties students' learning to accounts of what was happening in the classroom. "The all-to-familiar gulf between theoretical analyses and instructional practice is side stepped because theoretical insights about the means of supporting students’ learning in a particular domain are rooted in the practice of attempting to support that learning" (p 155). Third, the method is built around collaborating with teachers.

As for limitations, they first note that it may be hard to track student learning with this method when the classroom is traditional and the student thinking is not apparent. "A second limitation of the type of analysis we have illustrated is that its treatment
of social context is restricted to norms and practices that are established in the
course of face-to-face classroom interactions" (p 156). Finally, their method is limited because it can't be proceduralized since it deals not only with observable behavior but with "mathematical meaning" (p 157).

Our article: I think our article has a case of "usefulness" to make, but I don't think we will present it in such an obvious way. We will have our results, and we will have the affordances of the method, but it doesn't feel as strong to frame it as "usefulness" in our case. With regard to limitations, I think it will be valuable for us to present them clearly. First of all there are the limitations that are selected by the method because we are not trying to analyze everything in the discourse. But then there are also the limitations even within what we are trying to do (e.g., an issue of exposed student thinking as in Cobb, or the difficulty in picking up all the intertextual factors, etc.).


CONCLUSION (pp 157-158)

"Throughout this article, we have stressed that our overall goal is to be increasingly effective in developing instructional designs that support student’s mathematical learning" (p 157) ... "As we clarified, we find it essential to focus on the particular ways in which individual students are reasoning when we make instructional decisions in the classroom. In addition, we illustrated how the analysis of individual students’ reasoning can lead to conjectures about how we can improve our instructional designs" (pp 157-158) ... "Given these considerations, the methodology we have presented may be best viewed as a report from the field rather than a contribution to the ongoing debate between adherents of situated cognition and those who subscribe to more standard psychological paradigms. For us, the methodology is nothing more than a potentially revisable solution to the concrete problems and issues that we have encountered while experimenting in classrooms. We can therefore readily accept that alternative methodologies may be more appropriate for other purposes. We leave it to the reader to judge whether aspects of the analytical approach we have described are relevant to the problems of interest to them. In doing so, we extend the view of implementation as an idea-driven adaptation to our fellow researchers as well as to the teachers with whom we collaborate" (p 158).

Our article: They conclude by recapping the important decisions that they had to make with regard to the methodology and pointing out that other methodologies may also be appropriate, but there's has something significant to offer. I think conclusions are quite specific for articles, so ours may take on a different flavor. But this one is something to keep in mind.